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G = Q32⋊C2  order 64 = 26

2nd semidirect product of Q32 and C2 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.4D4, Q32⋊2C2, C4.15D8, C16.C22, SD32⋊2C2, C22.6D8, M5(2)⋊2C2, C8.11C23, D8.3C22, Q16.3C22, C4○D8.4C2, C4.12(C2×D4), C2.17(C2×D8), (C2×C4).48D4, (C2×Q16)⋊10C2, (C2×C8).24C22, 2-Sylow(ASigmaL(2,49)), SmallGroup(64,191)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C8 — Q32⋊C2
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×Q16 — Q32⋊C2
Lower central
C1 — C2 — C4 — C8 — Q32⋊C2
Upper central
C1 — C2 — C2×C4 — C2×C8 — Q32⋊C2
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — Q32⋊C2

Generators and relations for Q32⋊C2
 G = < a,b,c | a16=c2=1, b2=a8, bab-1=a-1, cac=a9, bc=cb >

C1
C2
2C2
8C2
C4
C4
C22
4C4
4C4
4C22
4C4
C8
C2×C4
C8
2D4
2Q8
2Q8
2Q8
4C2×C4
4Q8
4D4
4C2×C4
C16
C16
C2×C8
Q16
D8
Q16
Q16
2SD16
2C2×Q8
2C4○D4
2Q16
Q32
SD32
C4○D8
M5(2)
C2×Q16
Q32
SD32
Q32⋊C2

Character table of Q32⋊C2

 class 12A2B2C4A4B4C4D4E8A8B8C16A16B16C16D
 size 1128228882244444
ρ11111111111111111    trivial
ρ211-1-11-111-111-1-11-11    linear of order 2
ρ3111-111-1-1-11111111    linear of order 2
ρ411-111-1-1-1111-1-11-11    linear of order 2
ρ511-1-11-11-1111-11-11-1    linear of order 2
ρ61111111-1-1111-1-1-1-1    linear of order 2
ρ711-111-1-11-111-11-11-1    linear of order 2
ρ8111-111-111111-1-1-1-1    linear of order 2
ρ922-202-2000-2-220000    orthogonal lifted from D4
ρ10222022000-2-2-20000    orthogonal lifted from D4
ρ1122-20-22000000-√2√2√2-√2    orthogonal lifted from D8
ρ122220-2-2000000-√2-√2√2√2    orthogonal lifted from D8
ρ1322-20-22000000√2-√2-√2√2    orthogonal lifted from D8
ρ142220-2-2000000√2√2-√2-√2    orthogonal lifted from D8
ρ154-40000000-2√22√200000    symplectic faithful, Schur index 2
ρ164-400000002√2-2√200000    symplectic faithful, Schur index 2

Smallest permutation representation of Q32⋊C2
►On 32 points
Generators in S32
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)

(1 19 9 27)(2 18 10 26)(3 17 11 25)(4 32 12 24)(5 31 13 23)(6 30 14 22)(7 29 15 21)(8 28 16 20)

(1 26)(2 19)(3 28)(4 21)(5 30)(6 23)(7 32)(8 25)(9 18)(10 27)(11 20)(12 29)(13 22)(14 31)(15 24)(16 17)
 
G:=sub<Sym(32)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,19,9,27)(2,18,10,26)(3,17,11,25)(4,32,12,24)(5,31,13,23)(6,30,14,22)(7,29,15,21)(8,28,16,20), (1,26)(2,19)(3,28)(4,21)(5,30)(6,23)(7,32)(8,25)(9,18)(10,27)(11,20)(12,29)(13,22)(14,31)(15,24)(16,17)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,19,9,27)(2,18,10,26)(3,17,11,25)(4,32,12,24)(5,31,13,23)(6,30,14,22)(7,29,15,21)(8,28,16,20), (1,26)(2,19)(3,28)(4,21)(5,30)(6,23)(7,32)(8,25)(9,18)(10,27)(11,20)(12,29)(13,22)(14,31)(15,24)(16,17) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,19,9,27),(2,18,10,26),(3,17,11,25),(4,32,12,24),(5,31,13,23),(6,30,14,22),(7,29,15,21),(8,28,16,20)], [(1,26),(2,19),(3,28),(4,21),(5,30),(6,23),(7,32),(8,25),(9,18),(10,27),(11,20),(12,29),(13,22),(14,31),(15,24),(16,17)]])
 

Q32⋊C2 is a maximal subgroup of
 Q32⋊C4
 D8.D2p: D8.D4  D8.3D4  D8.12D4  C8.3D8  C8.5D8  SD32⋊S3  D8.9D6  SD32⋊D5 ...
 D2p.D8: D4○SD32  Q8○D16  Q32⋊S3  Q32⋊D5  Q32⋊D7 ...
 C8p.C23: D16⋊C22  C16.D6  C24.27C23  C16.D10  Q16.D10  C16.D14  Q16.D14 ...
Q32⋊C2 is a maximal quotient of
 C23.39D8  C23.41D8  M5(2)⋊1C4  SD32⋊3C4  Q32⋊4C4  Q16⋊7D4  C16⋊2D4  D8⋊1Q8  Q16⋊Q8  Q16.Q8  C22.D16  C23.50D8  C23.20D8  C8.12SD16  C8.14SD16  C16⋊Q8
 C16.D2p: C16.D4  C8.7D8  C16.D6  SD32⋊S3  Q32⋊S3  C16.D10  SD32⋊D5  Q32⋊D5 ...
 D8.D2p: D8.10D4  D8.4D4  D8.9D6  C40.31C23  C56.31C23 ...
 Q16.D2p: Q16.8D4  Q16.4D4  Q16.5D4  C24.27C23  Q16.D10  Q16.D14 ...

Matrix representation of Q32⋊C2 ►in GL4(𝔽7) generated by

2354
0321
2533
4446
,
3532
1442
1636
5524
,
6014
0545
0455
0665
G:=sub<GL(4,GF(7))| [2,0,2,4,3,3,5,4,5,2,3,4,4,1,3,6],[3,1,1,5,5,4,6,5,3,4,3,2,2,2,6,4],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5] >;
 

Q32⋊C2 in GAP, Magma, Sage, TeX 

Q_{32}\rtimes C_2
 
% in TeX
 
G:=Group("Q32:C2");
 
// GroupNames label
 
G:=SmallGroup(64,191);
 
// by ID
 
G=gap.SmallGroup(64,191);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,-2,-2,121,199,650,579,297,165,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=c^2=1,b^2=a^8,b*a*b^-1=a^-1,c*a*c=a^9,b*c=c*b>;
 
// generators/relations
 

Export

Subgroup lattice of Q32⋊C2 in TeX
Character table of Q32⋊C2 in TeX

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